RSA is not an alternative or replacement for DES; rather it supplements
DES (or any other fast bulk encryption cipher) and is used together
with DES in a secure communications environment. (Note: for an
explanation of DES, see Question 3.5.1)

RSA allows two important functions not provided by DES: secure key
exchange without prior exchange of secrets, and digital signatures. For
encrypting messages, RSA and DES are usually combined as follows: first
the message is encrypted with a random DES key, and then, before being
sent over an insecure communications channel, the DES key is encrypted
with RSA. Together, the DES-encrypted message and the RSA-encrypted DES
key are sent. This protocol is known as an RSA digital envelope.

One may wonder, why not just use RSA to encrypt the whole message and
not use DES at all? Although this may be fine for small messages, DES
(or another cipher) is preferable for larger messages because it is
much faster than RSA (see Question 3.2.3).

In some situations, RSA is not necessary and DES alone is sufficient.
This includes multi-user environments where secure DES-key agreement
can take place, for example by the two parties meeting in private.
Also, RSA is usually not necessary in a single-user environment; for
example, if you want to keep your personal files encrypted, just do so
with DES using, say, your personal password as the DES key. RSA, and
public-key cryptography in general, is best suited for a multi-user
environment. Also, any system in which digital signatures are desired
needs RSA or some other public-key system.